Can an object ever "own" an AN node tree or are the trees always independent?
I'm very new to AN so this might not be accurate.

I think the answer is no, you can't associate an animation nodes graph with an object; it's part of the scene. I think this is very much by design however, as you can have one node graph control many objects, even selecting an entire collection of objects at once.

This doesn't mean you can't have the features you want! Because AN is open source, you can always visit the project on GitHub, and under the "Issues" tab either request new features, or ask for further explanation about the design of the project.